The Benefits of Miscommunication Web

Consider the problem stated at the beginning of section 4.5, which involves choosing the distribution of posteriors F that maximizes (1 − α) 1 0 w(π) 1 − π 1 − π0 dF (π) expected surplus claimed by L +α 1 0 w(π) π π0 dF (π) expected surplus claimed by H −c 1 0 w(π) π π0 − 1 − π 1 − π0 dF (π) gap between expected surplus claimed by L and H 1 0 w(π) π π 0 − 1 − π 1 − π 0 dF (π) ≤ C(¯ m|L). To make further progress without imposing additional assumptions on cost functions, I decompose the problem into two steps. In the first step, I solve the problem for a given value of the gap between the expected surplus claimed by L and H, x = 1 0 w(π) π π 0 − 1−π 1−π 0 dF π (π) ∈ [0, w(1) − w(0)], henceforth referred to as the surplus gap, and derive the optimal distribution of posteriors conditional on x, denoted F * x. Note that the size of the surplus gap x is a measure of information transmission and thus the Lagrange multiplier associated with the constraint for x can be interpreted as the shadow value of information transmission. In the second step, I use F * x to determine the optimal surplus gap x * ∈ [0, min{w(1) − w(0), C(¯ m|L)}] that maximizes social welfare and infer the optimal distribution of posteriors F * = F * x * .